### Abstract

Topological complexity TC (B) of a space B is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version TC^{M} (B) of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities TC (B) = cat_{B}^{*} (d (B)) + 1 and TC^{M} (B) = cat_{B}^{B} (d (B)) + 1, where d (B) = B × B is a fibrewise pointed space over B whose projection and section are given by p_{d (B)} = pr_{2} : B × B → B the canonical projection to the second factor and s_{d (B)} = Δ_{B} : B → B × B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres.

Original language | English |
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Pages (from-to) | 10-21 |

Number of pages | 12 |

Journal | Topology and its Applications |

Volume | 157 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*157*(1), 10-21. https://doi.org/10.1016/j.topol.2009.04.056

**Topological complexity is a fibrewise L-S category.** / Iwase, Norio; Sakai, Michihiro.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 157, no. 1, pp. 10-21. https://doi.org/10.1016/j.topol.2009.04.056

}

TY - JOUR

T1 - Topological complexity is a fibrewise L-S category

AU - Iwase, Norio

AU - Sakai, Michihiro

PY - 2010/1/1

Y1 - 2010/1/1

N2 - Topological complexity TC (B) of a space B is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version TCM (B) of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities TC (B) = catB* (d (B)) + 1 and TCM (B) = catBB (d (B)) + 1, where d (B) = B × B is a fibrewise pointed space over B whose projection and section are given by pd (B) = pr2 : B × B → B the canonical projection to the second factor and sd (B) = ΔB : B → B × B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres.

AB - Topological complexity TC (B) of a space B is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version TCM (B) of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities TC (B) = catB* (d (B)) + 1 and TCM (B) = catBB (d (B)) + 1, where d (B) = B × B is a fibrewise pointed space over B whose projection and section are given by pd (B) = pr2 : B × B → B the canonical projection to the second factor and sd (B) = ΔB : B → B × B the diagonal. In addition, our method in studying fibrewise L-S category is able to treat a fibrewise space with singular fibres.

UR - http://www.scopus.com/inward/record.url?scp=70350164764&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350164764&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2009.04.056

DO - 10.1016/j.topol.2009.04.056

M3 - Article

AN - SCOPUS:70350164764

VL - 157

SP - 10

EP - 21

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 1

ER -